ON DIRECTED 2-FACTORS IN DIGRAPHS AND 2-FACTORS CONTAINING PERFECT MATCHINGS IN BIPARTITE GRAPHS.

In this paper, we give the following result: If D is a digraph of order n, and if d+ D(u)+d D(v) n for every two distinct vertices u and v with (u; v) =2 A(D), then D has a directed 2-factor with exactly k directed cycles of length at least 3, where n 12k+3. This result is equivalent to the following result: If G is a balanced bipartite graph of order 2n with partite sets X and Y , and if dG(x) + dG(y) n + 2 for every two vertices x 2 X and y 2 Y with xy =2 E(G), then for every perfect matching M, G has a 2-factor with exactly k cycles of length at least 6 containing every edge of M, where n 12k + 3. These results are generalizations of theorems concerning Hamilton cycles due to Woodall [Proc. Lond. Math. Soc., 24 (1972), pp. 739{755] and Las Vergnas [Problemes de couplages et problemes hamiltoniens en theorie des graphes, Ph.D. thesis, University of Paris, 1972], respectively. [ABSTRACT FROM AUTHOR]